Definition 1.1 ([1]})
Let \(f(x)\)  be a function from \(\mathbb {F}_{2^{n}}\)  to itself. For a fixed \(x_{0}\in \mathbb {F}_{2^{n}}\) , then we call the function \(f(x)\)  is \(x_{0}\) -APN (or partially APN) if all the points \(x, y\)  satisfying \(f(x_{0}) + f(x) + f(y) + f(x_{0} + x + y) = 0\)  belong to the curve \((x_{0} + x)(x_{0} + y)(x + y) = 0\) .

